in traditional Euclidean geometry we work with points,
lines, areas and volumes

Euclidean dimensions (E) are all positive whole
numbers

the Euclidean dimension represents the number of
coordinates necessary to define a point

to specify any point on a profile requires two
coordinates, thus a profile has a Euclidean
dimension of two

to define a point on a surface requires three
dimensions, therefore a surface has a Euclidean
dimension of three

closely allied with Euclidean dimensions are the
topological dimensions (DT) of phenomena

on a flat piece of paper (which has a Euclidean
dimension of 2) you can draw a two-dimensional
figure (DT= 2), a one-dimensional line (DT= 1), and
a zero-dimensional point (DT= 0) (compare 0-cell, 1-
cell and 2-cell notation)

in fractal geometry we work with points, lines, areas and
volumes, but instead of restricting ourselves to integer
dimensions, we allow the fractal dimension (D) to be any
real number

the limits on this real number are that it must be
at least equal to the topological dimension of the
phenomenon, and at most equal to the Euclidean
dimension (i.e., 0&AMPLT=DT&AMPLT=D&AMPLT=E)

a line drawn on a piece of paper can have a fractal
dimension anywhere from one to two

the term fractals is derived from the same Latin root
[fractus] as fractions; therefore: fractional dimensions

the fractal dimension summarizes the degree of complexity
of the phenomenon, the degree of its 'space-filling
capability'

overhead - Lines of different fractal dimensions

straight line will have equivalent topological and
fractal dimensions of 1

slightly curved line will still have a topological
dimension of 1, but a fractal dimension slightly
greater than 1

highly curved line (DT= 1) will have a much higher
fractal dimension

line which completely 'fills in' the page will have
a fractal dimension of 2

many natural cartographic lines have fractal
dimensions between 1.15 and 1.30

a surface can have a fractal dimension anywhere from
2 (perfectly flat) to 3 (completely space-filing)

fractal dimension indicates how measures of the object
change with generalization

e.g. a line with a low fractal dimension (straight
line) keeps the same length as scale changes

a line with fractal dimension 1.5 loses length
rapidly if it is generalized

topological dimension tells us little about how shapes
differ

e.g. all coastlines have the same topological
dimension

however, sections of many coastlines have been found
to have very different fractal dimensions

fractal dimension quantifies the metric information in
lines and surfaces in a new and unique manner

an example of how to determine the fractal dimension of a
cartographic line:
1. step a pair of dividers (step size s1) along
the line; say it takes n1 steps to span the line
2. the length of the line is equal to s1n1
3. repeat the process, but decrease the step size
(to s2); it now takes n2 steps to span the line
4. the length of the line is now s2n2
5. the fractal dimension can be calculated as:

the best method for generalizing lines may be
that method which best retains the fractal
dimension of the line
3. displaying lines at a scale greater than that
at which the line was collected

introduce additional "information", by adding
artificial detail to the line, detail which is
a function of the fractal dimension of the
original line);
4. incorporating the fractal dimension into
traditional cartometry measures

indicates that some aspect of a process or phenomenon is
invariant under scale-changing transformations, such as
simple zooming in or out

can be expressed in two ways:

overhead - Self-similarity
1. geometric self-similarity, in which there is
strict equality between the large and small scales

not found in natural phenomena

the Morton order, quadtrees use this idea in
replicating the same pattern at every level
2. statistical self-similarity, in which the
equality is expressed in terms of probability
distributions

this type of (random) self-similarity is the
more common, and is the type found in many
natural phenomena, such as coastlines, soil pH
profiles, river networks (Burrough, 1981;
Peitgen and Saupe, 1988; etc.)

the simplest test of self-similarity is visual

if a phenomenon is self-similar, any part of it, if
suitably enlarged, should be indistinguishable from
the whole or from any other part

if a natural scene is self-similar, it should be
impossible to determine its scale

e.g. it should be impossible to tell whether a
picture of self-similar topography shows a
mountain range or a small hill - there are no
visual cues as to the picture's scale

since many scale cues are cultural, geological or
geomorphological, self-similar topographies are most
common on lunar or recent volcanic landscapes

not necessarily equivalent to self-similarity, although
the two terms are often used interchangably in the
literature

consider a landscape, as represented by a surface and a
contour map

on the contour map (coordinates in 2 dimensions
only) the axes can be switched without fundamentally
changing the characteristics of the landscape, i.e.
the characteristics of the contour lines

contour lines are therefore examples of simple
scaling fractals

in the case of the surface, with coordinates in 3
dimensions, we cannot interchange the z axes with
either of the x or y axes without fundamentally
altering the characteristics of the landscape

since the z axis has a different scaling
parameter than the x or y axes, a three-
dimensional representation of the Earth's
surface is therefor an example of a non-uniform
(or multiple) scaling representation

shapes that are statistically invariant under
transformations that scale different coordinates by
different amounts are known as self-affine shapes
(Peitgen and Saupe, 1988)

the Earth's surface is an example of a self-affine
fractal, but it is not an example of a self-similar
fractal

contour lines, which represent horizontal cross-
sections of the land surface, are examples of
statistically self-similar scaling phenomenon
(because the contour has a constant z value)

because the land surface is self-affine and not self-
similar, those techniques which determine the fractal
dimension of the land surface itself produce values which
are different than the values produced by those
techniques which determine the fractal dimension of the
contours derived from that land surface

scale, through its relationships with generalization and
resolution, significantly influences length and area
measurements

problems in estimating line lengths, areas, and point
characteristics can be related to the phenomenon's
fractal dimension (Goodchild, 1980)

estimates of area are frequently based on pixel counts,
especially in raster-based systems

the error in the area estimate is a function of the
number of pixels cut by the boundary of the object

boundaries with a fractal dimension greater than one
will appear more complex as the pixel size decreases
(as the resolution increases)

the more contorted the boundary, or the higher its
dimension, the less rapid the increase in error with
cell size

diagram

error in a pixel-based area estimate will also be a
function of how the phenomenon is distributed about
the landscape: the error in area associated with a
highly compact phenomenon will be much less than the
error in area associated with a widely dispersed,
patchy phenomenon

Goodchild and Mark (1987, p. 268) show that:

the standard error as a percentage of the area
estimate is proportional to a(1-D/4) where a is
the area of a pixel and D is the fractal
dimension of the boundary

standard error will thus depend on a1/2 for
highly scattered phenomenon and a3/4 for
single, circular patches with smooth boundaries

Only a very small portion of the literature is presented
here. For further references you should refer to the
Goodchild and Mark (1987) paper; recent issues of Water
Resources Research and Science also contain relevant papers

1. Although fractal concepts are important in understanding
the error associated with pixel-based area estimates, little
has been said about the relationship between fractals and
area estimates obtained from vector-based systems. Why?
(i.e., would the area of an enclosed figure change
significantly? It is expected that the area shouldn't
change significantly, as the self-similar detail should
increase the area as much as it decreases the area.)

2. Define "fractal". Include in your description terms such
as scale dependency, self-similarity and scaling.

3. Discuss some of the ways in which fractals have changed
our way of looking at phenomena. Based on your readings,
provide examples from a variety of fields.

4. Theoretically, fractal behavior applies to a phenomenon
across all scales. Practically, of course, there are limits
to the application of self-similarity to natural phenomena.
Where do you think some of these limits occur? (i.e.,
between what scales do you think portions of coastlines, for
example, exhibit self-similar behavior.) What are the
implications with respect to the generalization of
cartogrpahic lines, if we observe definite limits to the
self-similar behavior of cartographic features?